Finite Group Databasebeta

← #275586

Subgroups — #275586

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29942C211yesmaximal, characteristic
2#27876C211yesmaximal, characteristic
3#22640C211yesmaximal, characteristic
4#21962C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10597C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#9622C211yesmaximal, characteristic, complemented
10#9612C211yesmaximal, characteristic, complemented
11#9575C211yesmaximal, characteristic, complemented
12#9568C211yesmaximal, characteristic, complemented
13#9127C211yesmaximal, characteristic
14#8830C211yesmaximal, characteristic
15#8728C211yesmaximal, characteristic
16C22×C4⋊C8C2211yescharacteristic
17#2857C2211yescharacteristic
18#2285C2211yescharacteristic
19#2283C2211yescharacteristic
20#2101—21no—
21#2085—21no—
22#2077—21no—
23#2077—21no—
24#2077—21no—
25#2075—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1987C2211yescharacteristic
30#1949—21no—
31#1949—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1924—21no—
41#1923—21no—
42#1919—21no—
43#1919—21no—
44#1844C2211yescharacteristic
45#1840—21no—
46#1834C2211yescharacteristic
47#1808—21no—
48#1807—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1768—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#389—21no—
101#389—21no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#383—21no—
106#382—21no—
107#382D411yescharacteristic
108#382D411yescharacteristic
109#381—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380D411yescharacteristic
115#380D411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121#376—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#334—22no—
147#334—22no—
148#333—22no—
149#333—22no—
150C22×Q8—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—24no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—81no—
162C22×C8—81no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C22⋊Q8—82no—
172#117—82no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204#107—81no—
205#107—81no—
206#107—81no—
207#106—42no—
208#106—42no—
209#106—81no—
210#106—81no—
211#106—81no—
212C4⋊C8—24no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8#4112yes—
226C4⋊C8#4112yes—
227C22⋊C8—22no—
228C22⋊C8—41no—
229C22⋊C8—81no—
230C22⋊C8—81no—
231#97—81no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—81no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24#12111yescharacteristic
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—81no—
253C2×Q8—81no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—84no—
314C2×C8#13512yes—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C22⋊C4—42no—
344C22⋊C4—81no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C422−1+411yescharacteristic
361C422−1+411yescharacteristic
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23#47112yes—
369C23C2×#12111yescharacteristic
370Q8—161no—
371Q8—161no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4#52511yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#54711yescharacteristic
420C2×C4#54711yescharacteristic
421C2×C4#55211yescharacteristic
422C2×C4#55211yescharacteristic
423C2×C4C2×#13511yescharacteristic
424C2×C4#57511yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57711yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C2×C4#57811yescharacteristic
431C8—22no—
432C8—41no—
433C8—41no—
434C8—41no—
435C8—42no—
436C8—42no—
437C8—81no—
438C8—81no—
439C8—81no—
440C8—81no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22#241311yescharacteristic
449C22#303511yescharacteristic
450C22#304611yescharacteristic
451C22#306011yescharacteristic
452C22#306311yescharacteristic
453C22C2×#47111yescharacteristic
454C22#306811yescharacteristic
455C4—21no—
456C4—21no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4#312111yescharacteristic
469C4#312311yescharacteristic
470C4#324211yescharacteristic
471C4#324411yescharacteristic
472C4#333711yescharacteristic
473C4#334211yescharacteristic
474C4#334911yescharacteristic
475C4#335011yescharacteristic
476C2—42no—
477C2#1922111yescharacteristic
478C2#1925311yescharacteristic
479C2#1954911yescharacteristic
480C2#1958811yescharacteristic
481C2#2458411yescharacteristic
482C2#2461911yescharacteristic
483C2#2462311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10656C2semidirect
6. #10653C2semidirect
7. #10597C2semidirect
8. #10589C2semidirect
9. #9622C2semidirect
10. #9612C2semidirect
11. #9575C2semidirect
12. #9568C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#299422
#278762
#226402
#219622
#106562
#106532
#105972
#105892
#96222
#96122
#95752
#95682
#91272
#88302
#87282